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CORRELATION FUNCTIONS OF CLASSICAL AND QUANTUM ARTIN SYSTEM DEFINED ON LOBACHEVSKY PLANE
H.Babujian, R.Poghossian, H.Poghosyan and G.Savvidyarxiv:1802.0454v2,arxiv:1808.02132v4
“Supersymmetries and Quantum Summetries”-Yerevan -29.08.2019,
THE CONTENTS
1.The motivation 2.The classical Artin system on Lobachevski
plane-Poincare metric. 3.The solution of the geodesic equations
and fundamental region. 4.Chaos in Artin system and exponential
instability; two point function. 5.The quantization of the Artin system.
THE CONTENTS 6. The quantum spectrum of the Artin
system in fundamental region; the Maas formula.
7. The two point function. 8. 4-point out of order correlation function
and scrambling time. 9. Conclusion. 10.References.
MOTIVATIONS. A. The experimental revolution-the enormous
improvements in experimental techniques it became feasible to control quantum systems with many degrees of freedom. An entirely new arena for the study of physics of interacting quantum many-body systems.
B. New mashines -Supercomputers and novel numerical techniques(tensor network , density matrix renormalization, Bethe Ansatz)
C. Understanding of the quantum mechanics has improved significantly since the of von Neumann!! The new mathematical methods.
MOTIVATIONS.
D. And finally –the problem of quantum gravity and black holes . There is the speculation that black holes should be most quantum chaotic system with maximal scrambling time t*=ß/2π .
t
ARTIN DYNAMICAL SYSTEM ON LOBACHEVSKI PLANE-POINCARE METRIC.
In 1924 Emil Artin introduced an example of Ergodic Dynamical system which is realized as geodesic flow on a compact surface of the ƑLobachevsky plane. The aim of this article was to construct an example of a dynamical system in which “almost all” geodesic trajectories are quasi-ergodic , meaning that all trajectories with the exception of measure zero , during their time evolution will approach infinitely close any point and given direction on surface . Let us consider ƑLobachevsky plane realized in the upper half-plane y>0 of the complex plane z=x+iy with
ARTIN SYSTEM
The Poincare metric which is given by the line element
22
22
)(Im
2
zzdzd
y
dydxds
The Lobachevsky plane is a surface a constant negative curvature , because its curvature Is equal to R=-2 and it is twice the Gaussian curvature K=-1. This metric has well known properties: it is invariant with respect to all linear substitution ,which form the group G of isometries of the Lobachevsky plane
ARTIN SYSTEM
zzzgw .
Where ,,,are real coefficients of the matrix g and the determinant of g is positive.
ARTIN SYSTEM
The equation for the geodesic lines on curved surface the form
02
2 dtdx
dtdx
kli
dtxd lki
Where kli are the Christoffer symbols of the Poincare metric
21yikikg
THE SOLUTION OF THE GEODESIC EQUATIONS
The geodesic equation takes the form
0)(1
)(1
02
222
2
2
2
dsdy
dsdx
ds
yd
dsdy
dsdx
dsxd
yy
y
and has two solutions
THE SOLUTION OF GEODESIC EQUATIONS
)exp()(,)(
)(),tanh()(
0
)cosh(0
ttyxtx
tytrxtx tr
First is the orthogonal semi-circles and second is perpendicular rays. Here
),0(),,(),,(0 rtx
THE SOLUTION OF THE GEODESIC EQUATION
Using this solution one can check the points on the geodesics curves move with unit velocity
1dtds
One can also observe that if we in the geodesic equation check the proper time,
tdxUEdt ))((2
Then the geodesic equation is coincide with equation of motion-the Newton equation.
In this case the potential is
21)(y
xU
THE FUNDAMENTAL REGION Ƒ In order to construct a compact surface on Ƒ
Lobachevsky plane , one can identify all points in upper half of the plane which are related to each other by the substitution g with integer coefficient and a unit determinant . These transformation form a modular group .Thus we consider two points z and w to be “identical” if
qpznmzw
THE FUNDAMENTAL REGION Ƒ With integers m,n,p,q constrained by the
condition mq-pn=1 which is unit determinant condition-w=dz ,det( d)=1. These elements d form discrete group SL(2,Z) which is discrete subgroup of the isometry transformation SL(2,R).The identification creates a regular tessellation of the Lobachevsky plane by congrouent hyperbolic triangles. The Lobachevsky plane is covered by the infinite-order triangular tiling.
THE FUNDAMENTAL REGION
THE FUNDAMENTAL REGION Ƒ One of these triangles can be chosen as a
fundamental region. That fundamental region of the modular group SL(2,Z),is the well Ƒ
known “modular triangle” consisting of those Points between lines x=- 1/2, and x==1/2
which lie outside the unit circle in the Fig.The modular triangle has two equal angles
and with the third one equal to zero,3
0
THE FUNDAMENTAL REGION Thus 3
2
The area of the fundamental region is finite and equal to
Inside the modular triangle there is exactly one Ƒrepresentative among all equivalent points of the Lobachevsky plane with exception of the points on the triangle edges are opposite to each other . These points should be identified to form a closed compact surface
3
F
THE FUNDAMENTAL REGION
By “gluing” the opposite edges of the mudular triangle together. The main goal of the construction is to consider now the behavior of the geodesic trajectories defined on the surface
constant negative curvature. In order to describe the behaviour of the geodesic
trajectories on the fundamental region one can use the knowledge of the geodesic trajectories on whole Lobachevsky plane. Arbitrary point on (x,y) and Ƒvelocity vector
CHAOS IN ARTIN SYSTEM)sin,(cos v
These are the coordinates of the phase space x,y, and theta belong to phase space M. and they uniquely determine the geodesic trajectory as the orthogonal circle K in the whole Lobachevsky plane. As trajectory “hits” the edges of the fundamental region and goes outside of it, one should apply the modular transformation to that parts of the circle K which are outside of in order to return Ƒthem back to the . That algorithm will define the whole Ƒtrajectory on fundamental region for arbitrary time . One should observe that this description of the trajectory on fundamental region is equivalent to the set of geodesic circles which appear
CHAOS IN ARTIN SYSTEM Under the action of the modular group on the initial circle K. One should join together the parts of the geodesic circles K’ which lie inside of into a unique continues trajectory on Ƒ Ƒ with boundaries . In this context the quasi-
ergodicity of the trajectory K on compact surface will mean that among all circle {K’} there are those which are approaching
arbitrary close to any given circle C!!
CHAOS IN ARTIN SYSTEM The geodesic trajectories are bounded to
propagate on the fundamental hyperbolic triangle.
The geodesic flow in this fundamental region represents one of the most chaotic dynamical systems with exponential instability of its trajectories , has mixing of all orders , Lebesgue spectrum and non-zero Kolmogorov entropy---Hedlund, Anosov, Hopf,Gelfand,Fomin,
Kolmogorov…
THE TWO POINT CORRELATION FUNCTIONS The earlier investigation of the correlation functions such a
Anosov systems was performed by- Pollicot , Moore , Dolgopyat…
using different approaches including Fourier series for the SL(2,R) group , the methods unitary representation theory. In our analyses we shall use the time evolution equation, the properties
of automorphic functions on and shall estimate a Ƒ decay exponent in terms of the phase space curvature and
the transformation properties of function . The correlation function can be defined as an integral over
a pair of functions/observables in which the first one is stationary and the second one evolves with geodesic flow:
THE TWO POINT FUNCTION
The f’ s are our physical observables and are the function on phase space . Here the measure and integral are
THE TWO POINT FUNCTION The SL(2,R)(SL(2,Z)) invariants fix the
functions/observables in upper half plane , these are [Gelfand ,Fomin] Poincare theta function of weight n -
THE TWO POINT FUNCTION
This theta function satisfy the condition
Still the last point , we should define the time evolution of the physical observables . The simplest evolution is
THE TWO POINT FUNCTION
The general evolution which map any geodesic to other and is the element of the isometry group SL(2,R) is
Now we can combine all these things together and after some calculation we will arrivethe final formula for the two point function when the time is
THE TWO POINT FUNCTION
In order to exclude the apparent singularity at the angle which solves the equation
If the surface has a negative curvature K
THE POINT FUNCTION
Then in last formula the exponential factor will take the form
Which shoes that when surface has larger negative curvature, then divergency of the trajectories is stronger.
THE QUANTUM ARTIN SYSTEM
This the action of the Artin system with equation of motion . We should notice the invariance of the action and equation of motions under time reparametrizations
)(tt
THE QUANTUM ARTIN SYSTEM
Presence of a local “gauge” symmetry indicates that we have a constrained dynamical system . One particularly convenient choice of gauge fixing specifying the time parameter t proportional to the proper time , is archived by imposing the condition
2
22
2y
yxH
THE QUANTUM ARTIN SYSTEM Where H is a constant . Defining the canonical momenta conjugate to the coordinates x , y as
22 , y
yyy
xx pp
We shall get the geodesic equations in the Hamiltionian form
yH
yx pp 2,0
THE QUANTUM ARTIN SYSTEM
Indeed , after defining the Hamiltonian as
)( 222
2
yxy ppH
The corresponding equation of motions will take the form as above , and advantageOf the gauge which we have,is that the Hamiltonian coincides with the constraint.Now it is fairly standard to quantize this Hamiltonian system we simply replace
yyxx ipip
,
THE QUANTUM ARTIN SYSTEM
And consider the Schrodinger equation
Ey
EH
yx )( 222
In this equation one easily recognises the Laplace operator with an extra minus sign,inPoincare metric. It is convenient to introduce parametrization E=s(1-s), as far E is real andSemi-positive and parametrization is symmetric with respect to s to 1-s and opposite .So , the the parameter s should be chosen within the range
],0[
]1,[
21
21
u
ius
s
THE QUANTUM ARTIN SYSTEM
One should impose the “periodic” boundary condition on the wave function with respect to the modular group
)()( zdczbaz
In order to have wave function which is propoerly defined on the fundamental region.Taking into account that transformation T:z to z+1 belongs to SL(2,Z),one has to impose the periodicity condition
)1()( zz
Thus we have a Fourier expansion
THE QUANTUM ARTIN SYSTEM
n
n inxyfyx )2exp()(),(
Inserting this to Schrodinger equation one get the solution for Fourier component
|)|2()(
0)()4)1(
21
2
2 22)(
ynKyyf
yfnss
sn
ndy
yfd n
THE QUANTUM ARTIN SYSTEM
For the case n=0 one simply gets and combining all together we get
ss ycycyf 1000 ')(
)2(exp|)|2(
'),(
0
100
21 inxynKcy
ycycyx
nn
sn
ss
THE QUANTUM ARTIN SYSTEM
Where all c’s should be defined such that the wave function will fulfill the boundary conditions
)()( zdczbaz
How we should solve this problem???H.Maas(1949)
THE QUANTUM ARTIN SYSTEM AND THEMAAS FORMULA
ss yz .)( 2
1
1)(
4
1)()1(
)2cos()2()(
)(
21
21
lsss
y
ssss
s
lxlyKl
yyz
THE MAAS FORMULA
Where we should define the several functions and element of SL(2,Z)
1,. bcadz dczbaz
)()(
)()2()(
.
nbaba
s
n
sss
THE CONTINUOUS AND DISCRETE SPECTRUM This wave function is well defined in the
complex s plane and has a simple pole at s=1. The physical continuous spectrum is defined by
241
21
21
)1(
],0[
,
],1,[
ussE
u
ius
s
THE DISCRETE SPECTRUM The wave function of the discrete spectrum
has a form
)}2sin(),2){cos(2()()(1
lxlxlyKynczniu
lln
And the coefficients c’s are not known analytically but where computed numerically for many values of n [D.A.Hejhal]. For the calculation they use the condition
241
1 )()(
nn
znn
uE
z
CORRELATION FUNCTIONS The Two –point Functions. Having explicit expressions of the wave functions
one can analyze the quantum-mechanical behavior of the correlation functions .
mnnmn
n
nBmmAnEtEEi
nHBiHtAiHtn
HBtAtD
,
2
|)0(||)0(|))((exp(
|)exp()0()exp()0()exp(|
)exp()0()(),(
CORRELATION FUNCTIONS
The energy eigenvalues are parametrized
ivm
iun
21
21
Thus we get for two point function
FiuivF iviu wdwBwzdzAz
utvuidvdutD
)()()()()()(
))()(exp(),(
21
21
21
21
0
24122
0
2
CORRELATION FUNCTIONS
Defining the basic matrix element as
)()(
)()()(
21
21
21
21 2
2
21
21
1
zAzdx
zdzAzA
iviu
x
y
dy
Fiviuuv
dudvBAutvuitD vuuv))()(exp(),( 24122
2
CORRELATION FUNCTIONS This expression is very convenient for the numerical calculation .One
should choose also appropriate observables A and B . The operator
seems very appropriate because ,the convergence of the integrals over fundamental region F will be much stronger.
It is expected that the two point correlation function should decay as
. In figure one can see the exponential decay of the two-point correlation function with time at different temperatures with the
2y
)exp(),(2 dttKtD
dt
THE TWO POINT FUNCTION
THE FOUR- POINT FUNCTION The out-of-time order four-point correlation
function is[ Maldacena , Shenker , Stenford]
dudvdldrBABAutrlvui
HBtABtAtD
rulrvluv))()(exp(
exp()0()()0()(),(
2412222
4
As well as the behavior of the commutator and square of commutator.
)exp()]0(),([ HBtA
THE FOUR POINT FUNCTION It was speculated in [Maldacena , Shenker ,
Stenford] , that the most important correlation functions including the races of the classical chaotic dynamics in quantum regime is
),('''),(''),('),(
)exp()]0(),([)(
4444
2
tDtDtDtD
HBtAtC
THE SCRAMLING TIME These formulas and the explicit form of the wave
functions allow to calculate the above correlation functions at least using numerical integration. Where we use so called plane wave approximation(we use only first two terms in wave function).Thus by performing numerical integration we observed that a two point correlation functions decays exponentially and that a four-point function demonstrate tendency to decay with a lower pace. With numerical data available to us it is impossible to estimate the scrambling time
2* t
THE SCRAMBLING TIME
Where the scrambling time is define as
)exp(1),(*04 ttftD
CONCLUSION
1.We recall the classical Artin system which is defined on the Lobachevsky plane .This system is one of the most chaotic dynamical system with exponential instability, has mixing of all order.
2.Using the Gelfand-Fomin differential geometry methods we calculate two point correlation function . This two point function decay exponentially.
3. We consider the quantization of the Artin system defined on the fundamental region of the Lobachevsky plane.
CONCLUSION 4.By performing a numerical integration we observed that the two
point correlation function of the quantum Artin system decays exponentially .
5.The four-point function demonstrates tendency to decay with a lower pace(in short time).
6. With a the numerical data available to us it is impossible to estimate scrambling time or to confirm its existence in the hyperbolic system , but qualitatively we observe a short time exponential decay of the out-of-time correlation function to almost zero value and then an essential increase with the subsequent large fluctuations.
7.In next we will continue to calculate the out –time- order four point correlation function in more details.
REFERENCES
1. Emil Artin . Ein mechanisches system mit quasiergodischen bahnen. E.Abh. Math. Semin. Univ. Hamburg.(1924),3 170 2. H. Maas, Uber eine neue Art von nichtanalytischen
automorphen Funktionen,Math.Ann.121,No.2(1949),141-183
3.L.D.Fadeev, Expansion in eigenfunctions of the Laplace operator on the fundamental domain of a discrete group on the Lobachevsky plane . Trans. Moscow Math.Soc.17(1967)357-386
4.D.A.Hejhal, The Selberg Trace Formula for PSL(2,R). Lecture Notes in Mathematics 548, Springer –Verlag Vol.1(1976)
